the state-variable filter

The state-variable filter is the Swiss Army knife of active filters: a single circuit that delivers low-pass, high-pass, and band-pass outputs all at once, from three separate pins, and whose cutoff and Q you can tune almost independently. It costs more op-amps than a Sallen-Key (usually three), but in return it is stable, tunable, and gives high, accurate Q that the simpler topologies struggle to reach.

It is built from a summing amplifier and two op-amp integrators in a loop, the same structure an analog computer uses to solve a second-order differential equation. Each integrator outputs a state of the system: tap one node and you get the high-pass response, the next gives band-pass, the last gives low-pass, simultaneously. The center frequency is set by the integrator resistors and capacitors (and can be tuned with a dual potentiometer without changing Q), while a separate resistor sets the Q on its own. Add a fourth op-amp to subtract band-pass from input and you also get a notch (band-stop) output, which is why it is the go-to for a deep, tunable notch.

State-variable filters are the workhorse for high-Q, precise, or tunable second-order sections, in graphic and parametric equalizers, lab instruments, tone controls, and stable notch filters. The honest tradeoffs: it uses three or four op-amps per second-order section (more cost, board space, and power), and it is still bounded by op-amp bandwidth and noise like any active filter. But when you need accuracy, a high stable Q, independent tuning, or several response types from one block, the extra op-amps are well spent.